Skip to main content

Binomial vs negative binomial differences

Short answer

Binomial has fixed trials and counts successes; negative binomial fixes successes and counts trials. Both use independent Bernoulli trials with constant p and are discrete.

Write the differences and similarities between Binomial and Negative Binomial Distribution.

Definitions

  • Binomial distribution: the number of successes X in a fixed number n of independent trials, each with success probability p. P(X = x) = C(n, x) pˣ qⁿ⁻ˣ, x = 0, 1, …, n; mean = np, variance = npq.
  • Negative binomial distribution: the number of trials X needed to get a fixed number r of successes. P(X = x) = C(x − 1, r − 1) pʳ qˣ⁻ʳ, x = r, r + 1, r + 2, …; mean = r/p, variance = rq/p².

Differences

BasisBinomialNegative binomial
Fixed quantityNumber of trials nNumber of successes r
Random variableNumber of successes in n trialsNumber of trials to get r successes
Range of X0 to n (finite)r, r + 1, … (infinite)
pmfC(n, x) pˣ qⁿ⁻ˣC(x − 1, r − 1) pʳ qˣ⁻ʳ
Meannpr/p
Variancenpq (always less than the mean)rq/p² (greater than the mean when q > p)
Last trialNo conditionThe last trial must be a success
ExampleHeads in 10 tossesTosses needed to get 3 heads

Similarities

  • Both are based on independent Bernoulli trials with only two outcomes (success or failure).
  • Both have a constant probability of success p in every trial, with q = 1 − p.
  • Both are discrete distributions.
  • When r = 1, the negative binomial becomes the geometric distribution, the waiting time for the first success in a Bernoulli process, which is the same process that underlies the binomial.
  • Both can be approximated by the Poisson or normal distribution under suitable conditions.

✓ Answer checked by Claude Sonnet 5.5

Related questions

All Probability and Statistics chapter 2 questions →